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Equal Diagonals (Posted on 2005-03-16) Difficulty: 3 of 5
If you were told to draw a rectangle along the lines of a sheet of graph paper such that its area is 40 squares, you could choose rectangles measuring 8x5, 10x4, 20x2 or 40x1.

For two of these, 8x5 or 10x4, you would find that you could draw a diagonal across the rectangle that would pass through exactly 12 squares.

What is the smallest number of squares that could be the area of three different rectangles whose diagonals pass through the same number of squares? How many squares does this diagonal pass through?

No Solution Yet Submitted by Sam    
Rating: 4.5000 (2 votes)

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Some Thoughts Solution | Comment 3 of 8 |

Assume that the phrase, three different rectangles, has been interpreted as three different set of rectangles and it does not mean three different shapes of rectangles. Assume that the phrase, diagonal passing through, in the question implies the three different ways of diagonals to be shown. Based on the above assumption, the answer for this question is 2x1 squares. Or in other words, there are three squares in total.


l                  l                     l

l                  l                     l

D------------- E----------------F For instance, if you have two set of rectangles as above, you could draw a straight line diagonally from C to D or A to F and that forms 2 rectangles.

The third rectangle could be shown vertically as follows:


I                    I 

l                     l


I                    I

I                    I


From the second set of rectangle, a line could be drawn from E to B and from F to A.  Thus, two diagonals can be formed in second set of rectangle.

Thus, it needs 2 squares to form a rectangle so as to have three different rectangles.


Edited on April 28, 2005, 6:09 am
  Posted by Jonathan Chang on 2005-04-28 05:58:28

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